Problem

Use the Rational Zero Theorem to list all possible rational zeros for the given function. \[ f(x)=x^{3}+15 x^{2}-14 x-14 \] Choose the answer below that lists all possible rational zeros. A. $-1,1,-14,14$ B. $-1,1,-\frac{1}{2}, \frac{1}{2},-\frac{1}{7}, \frac{1}{7},-\frac{1}{14}, \frac{1}{14}$ C. $-1,1,-2,2,-7,7,-14,14,-\frac{1}{2}, \frac{1}{2},-\frac{1}{7}, \frac{1}{7},-\frac{1}{14}, \frac{1}{14}$ D. $-1,1,-2,2,-7,7,-14,14$

Solution

Step 1 :The Rational Zero Theorem states that if a polynomial has a rational zero, then it must be a fraction in which the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient. In this case, the constant term is -14 and the leading coefficient is 1.

Step 2 :We need to find all the factors of -14 and 1, and then form all possible fractions with these factors.

Step 3 :The factors of -14 are -1, 1, -2, 2, -7, 7, -14, 14 and the factors of 1 are -1, 1.

Step 4 :Therefore, the possible rational zeros are all the fractions that can be formed with these factors.

Step 5 :The list of all possible rational zeros is \(\boxed{-1,1,-2,2,-7,7,-14,14}\).

From Solvely APP
Source: https://solvelyapp.com/problems/cJ8THjjwiI/

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